So far, we are
not finding winning strategies systematically. A mathematician may find
winning strategies for a game like chomp and relate all other games to
it. But many games can be won by the 2nd player, unlike chomp, so chomp
doesn’t seem general enough. We may generalize chomp so more games can
relate to it.
How?
One way is to increase the dimensions (play with cubes instead of
rectangles). But it is still a 1st-player-wins game (why?).
Another way is to allow the n by m rectangle to have infinite length or
width. This is more interesting. Notice the type of infinity matters;
suppose it is a 2 by infinity rectangle and suppose the 1st player’s
move so that at least one of the infinite row become finite, then the
2nd player wins by making a (finite) jug (exercise 3). So if that
infinity is the one that becomes finite no matter which box (and the
boxes to the right of it) you pick to remove, then 1st player loses
(define this type of infinity to be omega).
But if that infinity is not omega, but has 1 box to the right of the
omega many boxes, then 1st player wins (since he can make it into 2 by
omega with his move). This is omega+1, where 1 in the right means the
extra box is in the right. It would be different if the extra box is in
the left (1+omega=omega).
Generalizing games like nim this way (extending from natural numbers to
ordinals) yields nice results: a large class of games is equivalent to
nim.
Infinite games
We restricted attention to winning in a finite number of moves, but in
allowing game boards to have infinite size (such as playing chess on an
infinite board), it may be possible for a game with winning strategies
only after infinite number of moves.
N-player games
Activity
6
think of a way to generalize bridge-it if more than two people want to
play
[If there are
n players, you need some n-dimensional lattices (n of
them, each of which has all side lengths the same except one side is 1
unit longer) and then figure a way to arrange them nicely.]
Activity
7
How about Hex?
[For the
geometrically inclined, generalizing the hexagonal lattice to
3-dimension and higher may be a good idea. A different approach is to
find a way to play hex on a checker board, so we can generalize simply
by changing the board into an n-cube.] (Aside: the things said about
2-dimensional hex and Brouwer’s can be extended to n-dimension.)
Even though we haven’t learned much about how to win, let’s end here.
|
